1. Topology of Large Scale Structures Introduction, Theory and Progress Report 2004. 10. 28-29 Changbom Park (Korea Institute for Advanced Study) KIAS Workshop on Cosmology and Structure Formation

2. 1. Genus – A Measure of Topology Definition Definition G = # of holes - # of isolated regions G = # of holes - # of isolated regions = 1/4π· ∫ S κ dA (Gauss-Bonnet Theorem) = 1/4π· ∫ S κ dA (Gauss-Bonnet Theorem) [ex. G(sphere)=-1, G(torus)=0 ] [ex. G(sphere)=-1, G(torus)=0 ] : 2 holes – 1 body = +1

3. Why Topology? Why Topology? 1. Gaussianity of the primordial density field as predicted by many inflationary scenarios. 2. Topology of galaxy distribution at non-linear scales is sensitive to initial density power spectrum, cosmological parameters, and to galaxy formation mechanism. That is A. Initial density fluctuation  random phase  Gaussian (L scale) B. Gravitational Instability  G formation  non-Gaussian (NL scale) C. Non-gravitational effects of G formation  non-Gaussian (NL scale) GENUS as a PRECISION MEASURE

4. Genus - LS Galaxy Distribution Genus of iso-density contour surfaces in smoothed galaxy density distribution Genus of iso-density contour surfaces in smoothed galaxy density distribution as a function of density threshold level as a function of density threshold level (Weinberg, Gott & Melott 1987)

5. Genus - LS Galaxy Distribution Gaussian Field Gaussian Field Genus/unit volume g(ν) = A (1-ν 2 ) e - ν2/2 Genus/unit volume g(ν) = A (1-ν 2 ) e - ν2/2 where ν=(ρ- ρ b )/ ρ b σ & where ν=(ρ- ρ b )/ ρ b σ & A=1/(2π) 2 3/2 A=1/(2π) 2 3/2 if P(k)~k n, if P(k)~k n, A=[8√2π 2 R G 3 ] -1 A=[8√2π 2 R G 3 ] -1 * [(n+3)/3] 3/2 * [(n+3)/3] 3/2

6. Non-Gaussian Field (Toy models) Non-Gaussian Field (Toy models) Clusters Bubbles HDM (Weinberg, Gott & Melott 1987)

7. Genus-Related Statistics Amplitude drop R A Amplitude drop R A R A = A obs / A PS R A = A obs / A PS Shift parameter Δν Shift parameter Δν By fitting G obs ( ν) over –1< ν<1 By fitting G obs ( ν) over –1< ν<1 Asymmetry parameters A C & A V Asymmetry parameters A C & A V A = ∫ G obs ( ν) d ν/∫ G fit ( ν) d ν A = ∫ G obs ( ν) d ν/∫ G fit ( ν) d ν where intervals are where intervals are 1.2~2.2 ( A C ), -1.2~-2.2 ( A V ) 1.2~2.2 ( A C ), -1.2~-2.2 ( A V ) UZC+SSRS2

8. 2. History of Topology Study in Cosmology I. Early Works 1986: Hamilton, Gott, Weinberg; Gott, Melott, Dickinson 1986: Hamilton, Gott, Weinberg; Gott, Melott, Dickinson – smooth small-scale NL clustering to recover initial topology – smooth small-scale NL clustering to recover initial topology 1987-8: GWM, WGM, MWG, Gott et al. 1987-8: GWM, WGM, MWG, Gott et al. – cosmological & toy models. R G >3r c to recover initial topology – cosmological & toy models. R G >3r c to recover initial topology 1989: Gott et al. – observed galaxies, dwarfs, clusters 1989: Gott et al. – observed galaxies, dwarfs, clusters 1991: Park, Gott – gravitational & biasing effects 1991: Park, Gott – gravitational & biasing effects 1992: Weinberg, Cole – PS, initial skewness, biasing effects 1992: Weinberg, Cole – PS, initial skewness, biasing effects 1994: Matsubara – 2 nd order perturbation in weakly NL regime 1994: Matsubara – 2 nd order perturbation in weakly NL regime 1996: Matsubara – redshift space distortion in L regime 1996: Matsubara – redshift space distortion in L regime 1996: Matsubara, Suto – gravitational & z-space distortion 1996: Matsubara, Suto – gravitational & z-space distortion Etc…. Etc….

9. II. Recent Works 2000: Colley et al. – Simulation of SDSS 2000: Colley et al. – Simulation of SDSS 2001, 2003: Hikage, Taruya & Suto – dark halos (analytic & numerical) 2001, 2003: Hikage, Taruya & Suto – dark halos (analytic & numerical) 2003: Matsubara – 2 nd orber perturbation theory 2003: Matsubara – 2 nd orber perturbation theory [ Minkowski functionals (Mecke, Buchert & Wagner 1994; Schmalzing & Buchert 1997 etc.)] [ Minkowski functionals (Mecke, Buchert & Wagner 1994; Schmalzing & Buchert 1997 etc.)] III. 3D genus analysis of observational data 1989: Gott et al. - CfA 1 etc. 1992: Park, Gott, & da Costa - SSRS 1 1992: Moore et al. - IRAS QDOT 1994: Rhoads et al. - Abell Clusters 1994: Vogeley et al. - CfA 1+2 1997: Protogeros & Weinbergs - IRAS 1.2Jy 1998: Springel et al. - IRAS 1.2Jy 1998: Canavezes et al. - IRAS PSCz 2002: Hikage et al. - SDSS EDR 2003: Hikage et al. - SDSS LSS Sample 12 2004: Canavezes & Efstathious - 2dFRGS

10. IV. 2D Genus (LSS) 2D genus before SDSS 2D genus before SDSS Suggested by Melott (1987) Suggested by Melott (1987) Coles & Plionis (1991): Lick Galaxy Catalogue Coles & Plionis (1991): Lick Galaxy Catalogue Plionins, Valdarnini, & Coles (1992): Abell and ACO cluster catalogue Plionins, Valdarnini, & Coles (1992): Abell and ACO cluster catalogue Park et al. (1992): CfA Slice Park et al. (1992): CfA Slice Colley (2000): Simulated SDSS Colley (2000): Simulated SDSS Park, Gott, & Choi (2001): HDF Park, Gott, & Choi (2001): HDF Hoyle, Vogeley & Gott (2002): 2dFGRS Hoyle, Vogeley & Gott (2002): 2dFGRS 2D genus with SDSS 2D genus with SDSS Hoyle, Vogeley & Gott (2002): weak evidence for variation in the genus with galaxy type Hoyle, Vogeley & Gott (2002): weak evidence for variation in the genus with galaxy type

11. ΛCDM Simulation ΛCDM Simulation (Kim & Park 2004) PMTree code PMTree code (Dubinski, Kim, Park 2003) 2048 3 mesh 2048 3 mesh (initial condition) 2048 3 (8.6G) 2048 3 (8.6G) CDM particles 1024 & 5632 h -1 Mpc 1024 & 5632 h -1 Mpc size boxes 50 & 275 h -1 kpc 50 & 275 h -1 kpc force resolutions 3. Gravitational Evolution, Biasing, Redshift Space Distortion Effects on Topology (Park et al. 1994) (Tegmark et al. 2004)

12. Park, Kim & Gott (2004) Genus of matter distribution = A sim / A PS Amplitude drop R A = A sim / A PS Shift Δν # of Voids and Clusters A V & A C (at z = 0, 1, 2, 5, 8) 85 21085 210

13. Biasing  A V at small scales Redshift space distortion  small for A V

14. 3. Analytic Model Matsubara (1994,2003) : perturbation theory Matsubara(1996): linear theory of z-effects

15. Sloan Digital Sky Survey 1. Imaging of North Galactic Cap 2.5m APO telescope with a mosaic CCD camera u, g, r, i, z photometric bandpasses  selected for spectroscopy 2. Spectroscopy ~ 10 6 galaxies & 10 5 quasars with rms z-error ~ 30 km/s 3. Samples Main Galaxies: r Pet < 17.77 ; Quasars Luminous Red Galaxies (LRG): z 0.4 samples Korean Scientist Group (KSG) KIAS: Changbom Park & SNU: Myeong Lee, Myungshin Im KNU: Myeong-Gu Park & SU : Hwankyung Sung

16. As of Oct. 14, 2004

17. SDSS LSS Sample 14 in equatorial coordinate (314K galaxies)

18. SDSS LSS Sample 14 in survey coordinate (314K galaxies)

19. SDSS galaxies in region 1

20. SDSS galaxies in Region 1 (Park et al. 2004)

21. Genus Analysis of SDSS LSS Sample 14 Best sample to test Gaussianity of primordial fluctuation (randomness of quantum fluctuation)  Not yet! Structure formation mechanism  OK ! If Gaussian, G(ν) = A(1- ν 2 )exp(- ν 2 /2)G = # of holes - # of isolated regions SDSS : Large volume & dense sampling of galaxies

22. Volume-limited subsamples For scale dependence For luminosity dependence

23. Best Subset -20.21< M r <-21.59 160 < r < 314 h -1 Mpc 0.054 < z < 0.107 31,580 galaxies d = 6.3 h -1 Mpc

24. Scale Dependence Within a Subset : Same place(structure) & luminosity

25. Luminosity Dependence Each Subset : Same place(structure) luminosity smoothing L1: -20.5 ~ -22.5 L3: -19.0 ~ -21.0 L2: -19.5 ~ -21.5 (25524, 154~235 h -1 Mpc) 9 subsets with the same # of galaxies

26. YOU CAN SEE Few bright galaxies in under dense regions !

27. Luminosity bias: Park et al. (1994): nearly scale-independent biasing underdense regions lack bright galaxies

28. Internal Physical Parameters of Galaxies Collective Physical Properties of Galaxy Subsets Morphology, Surface Brightness, Luminosity, Velocity Dispersion, Color, Spectral Type, SFR, etc Galaxy Clustering Properties (z): Correlation Function, Power Spectrum, Count in Cell, Topology, etc Velocity Field Halo Mass Distribution, Luminosity Function (z), Color-Magnitude Relation, etc. Environment: Local Density Galaxy Biasing, Gaussianity of Initial Density Fluctuation, Ω m, σ 8, b, etc. Ω m, σ 8, b, etc. Galaxy Formation Different Tracers of Structure Formation: Galaxy, Cluster, Group, Void, Quasar, etc.

29. Conclusions 0. Wait for SDSS to finish for LS topology 1. Topology analysis does differentiate galaxy species Brighter : meat-ball topology, smaller voids Brighter : meat-ball topology, smaller voids Fainter :bubble topology, bigger voids Fainter :bubble topology, bigger voids 2. Topology changes below the characteristic magnitude M r* =-20.4+5 log h 3. A V < 1 (few & big voids) consistently at all scales < 10 h -1 Mpc independantly of L  Not gravitational evolution effects.  Not gravitational evolution effects. Existence of biasing ! Existence of biasing !